Quaternionic Linear Operators Research Articles

Spectrum and analytic functional calculus in real and quaternionic frameworks: An overview

<abstract><p>An approach to the elementary spectral theory for quaternionic linear operators was presented by the author in a recent paper, quoted and discussed in the Introduction, where, unlike in works by other authors, the construction of the analytic functional calculus used a Riesz-Dunford-Gelfand type kernel, and the spectra were defined in the complex plane. In fact, the present author regards the quaternionic linear operators as a special class of real linear operators, a point of view leading to a simpler and a more natural approach to them. The author's main results in this framework are summarized in the following, and other pertinent comments and remarks are also included in this text. In addition, a quaternionic joint spectrum for pairs of operators is discussed, and an analytic functional calculus which uses a Martinelli type kernel in two variables is recalled.</p></abstract>

Fredholm theory in quaternionic Banach algebras

Muraleetharan and Thirulogasanthar (J. Math. Phys. 2018;59(10):103506, 27p.) introduced the concept of the Calkin S-spectrum of a bounded quaternionic linear operator. The study of this spectrum is established using Fredholm operator theory. Motivated by this, we study the general framework of the Fredholm element with respect to a quaternionic Banach algebra homomorphism. First, we investigate the Fredholm S-spectrum of the sum of two elements in a quaternionic Banach algebra by means of the Fredholm S-spectrum of the two elements. Next, we prove a perturbation result on this spectrum. We also study the boundary S-spectrum. As an application, we investigate the Fredholm and Weyl S-spectra of bounded right quaternionic linear operators.

Jacobson’s Lemma in the ring of quaternionic linear operators

Abstract In the present paper, we study the Jacobson’s Lemma in the unital ring of all bounded right linear operators ℬ R (X) acting on a two-sided quaternionic Banach space X. In particular, let A, B ∈ ℬ R (X) and let q ∈ ℍ \ {0}, we prove that w(AB) \ {0} = w(BA) \ {0} where w belongs to the spherical spectrum, the spherical approximate point spectrum, the right spherical spectrum, the left spherical spectrum, the spherical point spectrum, the spherical residual spectrum and the spherical continuous spectrum. We also prove that the range of (AB)2 − 2Re(q)AB + |q|2 I is closed if and only if (BA)2 − 2Re(q)BA + |q|2 I has closed range. Finally, we show that (AB)2 − 2Re(q)AB + |q|2 I is Drazin invertible if and only if (BA)2 − 2Re(q)BA + |q|2 I is.

Sharper uncertainty principles in quaternionic Hilbert spaces

The uncertainty principle for quaternionic linear operators in quaternionic Hilbert spaces is established, which generalizes the result of Goh‐Micchelli. It turns out that there appears an additional term given by a commutator that reflects the feature of quaternions. The result is further strengthened when one operator is self‐adjoint, which extends under weaker conditions the uncertainty principle of Dang‐Deng‐Qian from complex numbers to quaternions. In particular, our results are applied to concrete settings related to quaternionic Fock spaces, quaternionic periodic functions, quaternion Fourier transforms, quaternion linear canonical transforms, and nonharmonic quaternion Fourier transforms.

The Cholewinski‐Fock space in the slice hyperholomorphic setting

Inspired from the Cholewinski approach, we investigate a family of Fock spaces in the quaternionic slice hyperholomorphic setting as well as some associated quaternionic linear operators. In a particular case, we reobtain the slice hyperholomorphic Fock space on quaternions.

A direct approach to the $S$-functional calculus for closed operators

The $S$-functional calculus for slice hyperholomorphic functions generalizes the Riesz-Dunford-functional calculus for holomorphic functions to quaternionic linear operators and to $n$-tuples of noncommuting operators. For an unbounded closed operator, it is defined, as in the classical case, using an appropriate transformation and the $S$-functional calculus for bounded operators. This is however only possible if the $S$-resolvent set of the operator contains a real point. In this paper, we define the $S$-functional calculus directly via a Cauchy integral, which allows us to consider also operators whose resolvent sets do not contain real points. We show that the product rule and the spectral mapping theorem also hold true with this definition and that the $S$-functional calculus is compatible with polynomials, although polynomials are not included in the set of admissible functions if the operator is unbounded. We also prove that the $S$-functional calculus is able to create spectral projections. In order to do this, we remove another assumption usually made: we do not assume that admissible functions are defined on slice domains. Instead, we also consider functions that are defined on not necessarily connected sets. This leads to an unexpected phenomenon: the $S$-functional calculi for left and right slice hyperholomorphic functions become inconsistent and give different operators for functions that are both left and right slice hyperholomorphic. We show that any such function is the sum of a locally constant and an intrinsic function. For intrinsic functions both functional calculi agree, but for locally constant functions they must give different operators unless any spectral projection commutes with arbitrary scalars.

Berezin transform of slice functions

In a recent paper, we introduced the Berezin transform of quaternionic linear operators and we obtained some properties in the setting of weighted Bergman spaces of slice hyperholomorphic functions. Here, we define the Berezin transform of slice functions, which corresponds to the case of a multiplication operator by certain quaternion-valued slice functions.

Schatten class and Berezin transform of quaternionic linear operators

In this paper, we introduce the Schatten class and the Berezin transform of quaternionic operators. The first topic is of great importance in operator theory, but it is also necessary to study the second one, which requires the notion of trace class operators, a particular case of the Schatten class. Regarding the Berezin transform, we give the general definition and properties. Then we concentrate on the setting of weighted Bergman spaces of slice hyperholomorphic functions. Our results are based on the S‐spectrum of quaternionic operators, which is the notion of spectrum that appears in the quaternionic version of the spectral theorem and in the quaternionic S‐functional calculus. Copyright © 2016 John Wiley & Sons, Ltd.

Julia theory for slice regular functions

The theory of slice regular functions is nowadays widely studied and has found elegant applications to functional calculus for quaternionic linear operators and Schur analysis. However, much less is known about their boundary behaviors. In this paper, we initiate the study of Julia theory for slice regular functions. More precisely, we establish the quaternionic versions of the Julia lemma, the Julia-Caratheodory theorem, the boundary Schwarz lemma, the Hopf lemma, and the Burns-Krantz rigidity theorem for slice regular self-mappings of the open unit ball B ⊂ H and of the right half-space H+. We provide an explicit example to show that in quaternionic Hopf lemma the slice derivative of a slice regular function f at the boundary fixed point may not be real, in contrast to the complex version. Our result implies that the commonly believed version of the Hopf lemma turns out to be totally wrong. This new quaternionic version of the Hopf lemma also improves Osserman estimate even in the complex setting and is essentially the first significant theorem belonging to the theory of quaternions itself other than the theory of complex analysis, since the result involves the Lie brackets which reflect the non-commutative feature of quaternions.

The Spectral Theorem for Unitary Operators Based on the S-Spectrum

The quaternionic spectral theorem has already been considered in the literature, see e.g. [22], [31], [32], however, except for the finite dimensional case in which the notion of spectrum is associated to an eigenvalue problem, see [21], it is not specified which notion of spectrum underlies the theorem. In this paper we prove the quaternionic spectral theorem for unitary operators using the $S$-spectrum. In the case of quaternionic matrices, the $S$-spectrum coincides with the right-spectrum and so our result recovers the well known theorem for matrices. The notion of $S$-spectrum is relatively new, see [17], and has been used for quaternionic linear operators, as well as for $n$-tuples of not necessarily commuting operators, to define and study a noncommutative versions of the Riesz-Dunford functional calculus. The main tools to prove the spectral theorem for unitary operators are the quaternionic version of Herglotz's theorem, which relies on the new notion of $q$-positive measure, and quaternionic spectral measures, which are related to the quaternionic Riesz projectors defined by means of the $S$-resolvent operator and the $S$-spectrum. The results in this paper restore the analogy with the complex case in which the classical notion of spectrum appears in the Riesz-Dunford functional calculus as well as in the spectral theorem.

Recent development of the theory of Slice regular functions

Theory of Slice regular functions is a natural generalization of complex analysis from complex numbers to quaternions. It was introduced initially by Gentili and Struppa in 2006, and has been extensively studied and has found its elegant applications to functional calculus for quaternionic linear operators, operator theory, Schur analysis and quaternionic quantum mechanics. Meanwhile, the notion of Slice regularity was also extended to functions of an octonionic variable and to the setting of Clifford algebras as well as to the setting of alternative real algebras. In this survey, we shall focus mainly on our recent results in the theory of Slice regular functions. Firstly, we establish the Borel-Caratheodory inequality and the sharp growth and distortion theorems for Slice regular extensions of normalized biholomorphic functions on the unit disc in the setting of quaternions. In addition, we study the boundary behavior of Slice regular functions and obtain the Julia lemma, the Julia-Caratheodory theorem as well as the boundary Schwarz lemma. In particular, we find that the boundary Schwarz lemma does not ensure the Slice derivative of a Slice regular self-mappings of the open unit ball B⊂H at its boundary fixed point to be necessarily a real number, in contrast to that in the complex case. Finally, we study theory of function spaces of Slice regular functions and establish the Forelli-Rudin type estimates.

Right eigenvalue equation in quaternionic quantum mechanics

We study the right eigenvalue equation for quaternionic and complex linear matrix operators defined in n-dimensional quaternionic vector spaces. For quaternionic linear operators the eigenvalue spectrum consists of n complex values. For these operators we give a necessary and sufficient condition for the diagonalization of their quaternionic matrix representations. Our discussion is also extended to complex linear operators, whose spectrum is characterized by 2n complex eigenvalues. We show that a consistent analysis of the eigenvalue problem for complex linear operators requires the choice of a complex geometry in defining inner products. Finally, we introduce some examples of the left eigenvalue equations and highlight the main difficulties in their solution.